Schol, Dennis

Vlasiou, Maria

Zwart, Bert

English

We prove that the scaled maximum steady-state waiting time and the scaled maximum steady-state queue length among $N$ $GI/GI/1$-queues in the $N$-server fork-join queue, converge to a normally distributed random variable as $N\to\infty$. The maximum steady-state waiting time in this queueing system scales around $\frac{1}{\gamma}\log N$, where $\gamma$ is determined by the cumulant generating function $\Lambda$ of the service distribution and solves the Cram\'er-Lundberg equation with stochastic service times and deterministic inter-arrival times. This value $\frac{1}{\gamma}\log N$ is reached at a certain hitting time. The number of arrivals until that hitting time satisfies the central limit theorem, with standard deviation $\frac{\sigma_A}{\sqrt{\Lambda'(\gamma)\gamma}}$. By using distributional Little's law, we can extend this result to the maximum queue length. Finally, we extend these results to a fork-join queue with different classes of servers

University of Twente Research Information

Extreme values for the waiting time in large fork-join queues

https://ris.utwente.nl/ws/files/320251569/2309.08373v1.pdf

Extreme values for the waiting time in large fork-join queues

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University of Twente Research Information